# Solution of y''-(1+x)y=0 using power series ..

• Jan 14th 2011, 07:09 AM
Liverpool
Solution of y''-(1+x)y=0 using power series ..
Hello

Obtain the solution of the following differential equation as a power series about the origin :

$\displaystyle y''-(1+x)y=0$

Solution:

Let $\displaystyle \displaystyle y=\sum_{n=0}^{\infty} a_n x^n$
then :

$\displaystyle \displaystyle y'=\sum_{n=1}^{\infty} n a_n x^{n-1}$

$\displaystyle \displaystyle y''=\sum_{n=2}^{\infty} n (n-1) a_n x^{n-2}$

Rewrite the equation as : $\displaystyle y''-y -xy=0$

and substitute the values of y & y'' gives:

$\displaystyle \displaystyle \sum_{n=2}^{\infty} n (n-1) a_n x^{n-2} - \sum_{n=0}^{\infty} a_n x^n - \sum_{n=0}^{\infty} a_n x^{n+1}$

or:

$\displaystyle \displaystyle \sum_{n=0}^{\infty} (n+2)(n+1) a_{n+2} x^n - \sum_{n=0}^{\infty} a_n x^n - \sum_{n=0}^{\infty} a_n x^{n+1}$

My problem is in the last summation
I want to make the summations start with the same n and the powers of the x's are n in order to get the recursive relation, but I can not do that .

the last summation makes my life hard

any help?
• Jan 14th 2011, 07:36 AM
Ackbeet
Try breaking off the first term of the last summand thus:

$\displaystyle \displaystyle \sum_{n=0}^{\infty} (n+2)(n+1) a_{n+2} x^n - \sum_{n=0}^{\infty} a_n x^n - a_{0}x-\sum_{n=1}^{\infty} a_n x^{n+1}=0,$ then re-indexing gives

$\displaystyle \displaystyle \sum_{n=0}^{\infty} (n+2)(n+1) a_{n+2} x^n - \sum_{n=0}^{\infty} a_n x^n - a_{0}x-\sum_{n=0}^{\infty} a_{n+1} x^{n+2}=0.$

You might need to break off other terms as well, in order to get all the powers of x to be the same. Those are your two tricks: re-indexing and breaking off terms. You should be able to get all the starting indices and powers of x to be the same in all three summations, which would enable you to combine them into one summation.
• Jan 14th 2011, 10:11 AM
HallsofIvy
I'm not sure why Ackbeet put the last sum to power $\displaystyle x^{n+2}$. I would think you would want them all as $\displaystyle x^n$ so you could combine the sums. One way of doing that is, recognizing that you want to make "$\displaystyle x^{n+1}$" look like "$\displaystyle x^j$", set j= n+1. Then n= j- 1 so that $\displaystyle a_n= a_{j-1}$. Also when n= 0, j= 1 so the sum $\displaystyle \sum_{n=0}^\infty a_nx^{n+1}$ becomes $\displaystyle \sum_{j= 1}^\infty a_{j-1}x^j$. Relabel "j" to be "n" and you have
$\displaystyle \displaystyle \sum_{n=0}^{\infty} (n+2)(n+1) a_{n+2} x^n - \sum_{n=0}^{\infty} a_n x^n - a_{0}x-\sum_{n=1}^{\infty} a_{n-1} x^{n}=0$
Since the last sum does not start until n= 1, as Ackbeet suggests, do n= 0 separately:
With n= 0,
$\displaystyle 2a_2- a_0= 0$
and then
$\displaystyle \displaystyle \sum_{n=1}^{\infty} (n+2)(n+1) a_{n+2} x^n - \sum_{n=1}^{\infty} a_n x^n - a_{0}x-\sum_{n=1}^{\infty} a_{n-1} x^{n}=0$
so that for each n, greater than or equal to 1,
$\displaystyle (n+2)(n+1)a_{n+2}- a_n- a_{n-1}= 0$
or
$\displaystyle a_{n+2}= \frac{a_n+ a_{n-1}}{(n+2)(n+1)}$.

Presumably, you have some "initial conditions" that will give you $\displaystyle a_0$ ($\displaystyle y(0)= a_0$) and $\displaystyle a_1$ ($\displaystyle y'(0)= a_1$). Then the first equation tells you that $\displaystyle a_2= a_0/2$ so that you have $\displaystyle a_0$, $\displaystyle a_1$, and $\displaystyle a_2$. Then you can find $\displaystyle a_3$ by taking n= 1 in the last equation, find $\displaystyle a_4$ by taking n= 2, etc. With luck, you might be able to find a general formula but that is rare.
• Jan 14th 2011, 10:54 AM
zzzoak
I may be wrong, but for n=1 (as there is term $\displaystyle -a_0x$) it must be
written separately

$\displaystyle n=0: \; \; 2a_2-a_0=0$

$\displaystyle n=1: \; \; 6a_3-a_1-2a_0=0$

$\displaystyle n>=2: \; \; (n+2)(n+1)a_{n+2}-a_n-a_{n-1}=0$
• Jan 14th 2011, 01:48 PM
HallsofIvy
The "$\displaystyle a_0x$" term is my mistake. Being basically lazy, I copied the LaTex version of the equation from Ackbeet's post but, being basically careless, I copied the wrong equation! I intended to copy the first equation before he had separated that part out. What I should have had was:
$\displaystyle \displaystyle \sum_{n=0}^{\infty} (n+2)(n+1) a_{n+2} x^n - \sum_{n=0}^{\infty} a_n x^n - \sum_{n=1}^{\infty} a_{n-1} x^{n}=0$

I just copied it again, but this time I took out the "$\displaystyle -a_0x$" part!
• Feb 9th 2011, 02:22 AM
Liverpool
Thanks all.